A gambler goes to bet. The dealer has 3 dice, which are fair, meaning that the chance that each face shows up is exactly 1/6.

The dealer says: "You can choose your bet on a number, any number from 1 to 6. Then I'll roll the 3 dice. If none show the number you bet, you'll lose $1. If one shows the number you bet, you'll win $1. If two or three dice show the number you bet, you'll win $3 or $5, respectively."

Is it a fair game?

Hint

What will happen if there are 6 gamblers, each of whom bet on a different number?Hide

I disargee as to being a fiar game, If only one person plays, which is the feel of the problem to start with. There are 216 possible out comes from throughing 3 dies. 1/216 of winning $5, 5/216 of winning $3, and 25/216 of winning $1, but 185/216 of losing $1. Which means on average you lose $0.648 a game, you are better off playing cratchers.

No not fair. Every good game favours the house. If it doesn't no matter how fun the game is no one gets to play it long. Every game has upkeep costs. With no profitability the gambling will stop when the roller sstarves to death.

smurfdew... i disagree. kiss is actually right. great job kiss, I used another way to solve and was amazed that the expected value of a single player is actually zero (meaning, the game is fair).
To smurfdew...
There is 1 (1x1x1x(3C0)) way to win $5, 15 (1x1x5x(3C1)) ways to win $3 and 75 (1x5x5x(3C2)) ways to win $1. There only 125 (5x5x5x(3C3))ways to lose $1.

There are 216 possible rolls (6^3). Now if you would bet on 1 for all of these rolls you would only win $123. So you would lose (216-123=93) $93. Is this a fair game for the player I do not agree. You win 1 -$5 14 - $3 76 - $1.

Great brainteaser. I first did it without the hint via probability, but when I looked at the hint, boy did I feel defeated. I really hope someday I can be someone who can come up with that beautiful solution.

There are 216 possible rolls (6^3). Now if you would bet on 1 for all of these rolls you would only win $123. So you would lose (216-123=93) $93. Is this a fair game for the player I do not agree. You win 1 -$5 14 - $3 76 - $1.

I Agree it's unfair... the hint about dealer with 6 players is misleading; there is only a correct way to solve the puzzle, and is to compute the EXPECTED WIN of the player : if game is fair, it must be 0: a very simple computation returns (as already written by others) -37 cents as expected win : to say, if I AND THE DEALER will continue to play, in the long run I will lose my money by sure